Multipole representations, molecular charge distribution

Sokalski WA, Poirier RA (1983) Cumulative atomic multipole representation of the molecular charge distribution and its basis set dependence. Chem Phys Lett 98 86-92... 

The electrostatic interaction, which is defined as the classical Coulombic interaction between the undistorted charge distributions of the isolated molecules, is the easiest to derive from wavefunctions. When there is no overlap of the charge distributions of the molecules, all that is required is a representation of the molecular charge density. The traditional, and simplest, representation of the molecular charge distribution is in terms of the total multipole moments. The first nonvanishing multipole moment could often be derived from experi-... 

F. Vigne-Maeder and P. Claverie /. Ghent. Phys., 88, 4934 (1988). The Exact Multicenter Multipolar Part of a Molecular Charge-Distribution and Its Simplified Representations. R. J. Wheatley, Ghent. Phys. Lett., 208,159 (1993). A New Distributed Multipole Procedure for Linear Molecules. 

W. A. Sokalski, C/zem, Phys. Lerf., 98, 86 (1983). Cumulative Atomic Multipole Representation of the Molecular Charge Distribution and Its Basis Set Dependence. 

W. A. Sokalski and R. A, Poirer, Chem. Phys, Lett., 98, 86 (1983). Cumulative Atomic Multipole Representation of the Molecular Charge Distribution and Its Basis Set Dependence. D. E. Williams and D. J. Craycroft, J. Phys, Chem., 89, 1461 (1985). Estimation of Dimer Coulombic Intermolecular Energy and Site Charge Polarization by the Potential-derived Method. 

Segmental multipole moments contain extremely compact linearly scaling 0(N) representation of molecular charge distribution, which may otherwise occupy 0(N ) disk space in the form of electron density matrix. This feature should become more important in future, when advances in direct and linearly scaling algorithms will allow to produce routinely electron density matrices for very large molecular systems which could not be permanently stored. 

Indeed, this idea lies at the heart of the venerable notion of assigning partial charges to the atoms in a molecule [17]. One may think of this practice as a representation of the true charge distribution of the molecule by a series of distributed multipoles (in this case, limited to monopoles) at various sites, namely, atomic centers. Even if limited to monopoles, the act of spreading them out over the entire molecule is equivalent in some sense to simulation of high orders of molecular center-based multipoles. 

The distributed multipole analysis method of Stone and co-workers is similar in concept but is based on nonredundant spherical harmonic representation of the multipoles (recall that whereas there are six second moments, only five are independent). He initially places numerous site multipoles at centers of orbital overlap. The individual monopoles are spread out along the molecular axis, and are thought to represent the distribution of charge the site dipoles are also spread out along the bond axis. This very detailed description is simplified into a three-site model, which includes a site in the F—H bond. However, the multipole expansion does not converge well, especially for the bond center site. 

In the Buckingham-Fowler model, each monomer electric charge distribution is described by a set of point multipoles (charges, dipoles and quadru-poles) located on the atoms and, sometimes, additionally at bond midpoints. The values of the point multipoles are determined by the so-called distributed multipole analysis (DMA) of an ab initio wavefunction. This multicentric representation of the charge distribution shows superior convergence behaviour to the one-centre molecular multipoles when calculating the electrostatic potential around a molecule. 

If the charge distribution is described by a set of distributed multipoles, as described in Section 4.2.3, the coulombic contributions to the intermolecular potential energy are calculated as multipole-multipole terms. The main disadvantage of even a rigorous distributed multipole model is that such a representation is still very localized, so that coulombic energies miss a large part of the penetration contribution. For use in a complete representation of intermolecular interactions, the dispersion, polarization, and repulsion terms must be evaluated separately by some semi-empirical or fiilly empirical method, for example the approximate atom-atom formulations of equations 4.38. 39. This approach has been extensively exploited by S. L. Price and coworkers over the years, in applications to molecular crystals [48]. 

An alternative approach was to include explicit, higher order electrostatic moments in the pairwise interactions. This approach has not been extensively developed for use in molecular simulations because of the complex set of moments needed to obtain sensible results, particularly to mimic hydrogen bonding. A notable exception is the polarizable electropole model, which relies on a central polarizability as well as higher order moments to capture the electrostatic part of the interactions." The computational effort required for a multipole-based representation of the electrostatics is much greater than is involved in the use of distributed charges to represent the electrostatic interactions. If, on the other hand, the number of partial charge sites is substantially increased, a local expansion of multipole moments can become computationally economical. ... 

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